In Refs. [1]}, [2]}, [3]} on the
nonlocally interacting Brownian and Lévy bugs it was assumed
that the birth and death rates of the \(i\) -th individual are
given by Eqs. (REF ), (). In the case of
Brownian bugs, for small enough diffusion coefficient and large
enough \(\Delta _0\) , the occurrence of a periodic pattern
consisting of clusters that are arranged in a hexagonal lattice
was observed (see Fig. REF a-c)
[1]}, [2]}. For large values of the
diffusion coefficient such periodic pattern is replaced by a
more homogeneous distribution of bugs (Fig. REF d).
In the case of Lévy bugs, since the diffusion coefficient
(REF ) is infinite, one could expect that the spatial
distribution will not reveal a periodic pattern; however, as
shown in Ref. [3]}, for proper parameters
periodic cluster arrangements do indeed occur (see
Fig. REF ). The reason for the divergence of the diffusion
coefficient in the Lévy case is in the statistical weight of
large jumps. These large jumps have some influence on the
characteristics of the pattern formed, but the relevant
structure is ruled mainly by the interactions between
individuals. In the Lévy bug system however, at variance with
the Brownian case, even at small values of \(\kappa _\mu \)  there
are many solitary bugs appearing for short time periods in the
space between the periodically arranged clusters due to the
large jumps [3]}, c.f. Figs. REF a
and REF a. However, the periodicity of the pattern is of
the order of \(R\)  (the interaction range) in both systems, being
only slightly influenced by \(\kappa \)  or \(\kappa _\mu \)  and
\(\mu \) , as demonstrated in Refs. [1]}, [3]} through a mean-field theory calculation.
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